nLab wave vector

Contents

Context

Harmonic analysis

Idea
Definition
Related concepts
References

Idea

A wave vector is a vector that encodes wavelength and direction of a plane wave.

Definition

Let $n \in \mathbb{N}$ and write $\mathbb{R}^n$ the Cartesian space of dimension $n$ . Thinking of $\mathbb{R}^n$ as a vector space, then each point in it is a vector $\vec x \in \mathbb{R}^n$ and hence a smooth function $f \colon \mathbb{R}^n \to \mathbb{C}$ may be thought of as a function of these “position vectors”.

If $f$ is a function with rapidly decreasing partial derivatives, then its Fourier transform $\hat f \;\colon\; \mathbb{R}^n \to \mathbb{C}$ exists. By the Fourier inversion theorem, this function is such that it expresses $f$ as a superposition of “plane wave” functions $\vec x \mapsto e^{2\pi i \vec x \cdot \vec k}$ as

f(\vec x) \;=\; \underset{\vec k \in \mathbb{R}^n}{\int} \hat f(k) \, e^{2 \pi i \vec k \cdot \vec x} \, d \vec k \,.

Here the vector $\vec k \in \mathbb{R}^n$ determines

the wavelength $\lambda \coloneqq 1/{\vert \vec k\vert}$ (the inverse of the norm of $\vec k$ );
the direction $\frac{\vec k}{{\vert \vec k\vert }} \in S(\mathbb{R}^n)$ (the corresponding unit vector in the unit sphere)

of the “plane wave” $\vec x \mapsto e^{2 \pi i \vec x \cdot \vec k}$ .

The product $2 \pi {\vert k \vert}$ is also called the wave number and $2 \pi k$ then the wave number vector. Beware that elsewhere the wave number vector is denoted “ $k$ ”, which makes the “wave vector” become $k / 2 \pi$ . (See e.g. Wikipedia, “Physics definition” as opposed to “Crystallography definition”.)

If here $\mathbb{R}^n \simeq \mathbb{R}^{p,1}$ is identified with Minkowski spacetime with canonical coordinates denoted $(x^0, x^1, \cdots, x^p)$ , then the 0-component of the wave vector

\nu \coloneqq k_0

is called the frequency of the corresponding plane wave (in the chosen coordinate system); this $\omega = 2 \pi \vu$ is the angular frequency.

plane waves on Minkowski spacetime

\array{ \mathbb{R}^{p,1} &\overset{\psi_k}{\longrightarrow}& \mathbb{C} \\ x &\mapsto& \exp\left( \, i k_\mu x^\mu \, \right) \\ (\vec x, x^0) &\mapsto& \exp\left( \, i \vec k \cdot \vec x + i k_0 x^0 \, \right) \\ (\vec x, c t) &\mapsto& \exp\left( \, i \vec k \cdot \vec x - i \omega t \, \right) }

symbol	name
$c$	speed of light
$\hbar$	Planck's constant
$\,$	$\,$
$m$	mass
$\frac{\hbar}{m c}$	Compton wavelength
$\,$	$\,$
$k$ , $\vec k$	wave vector
$\lambda = 2\pi/{\vert \vec k \vert}$	wave length
${\vert \vec k \vert} = 2\pi/\lambda$	wave number
$\omega \coloneqq k^0 c = -k_0 c = 2\pi \nu$	angular frequency
$\nu = \omega / 2 \pi$	frequency
$p = \hbar k$ , $\vec p = \hbar \vec k$	momentum
$E = \hbar \omega$	energy
$\omega(\vec k) = c \sqrt{ \vec k^2 + \left(\frac{m c}{\hbar}\right)^2 }$	Klein-Gordon dispersion relation
$E(\vec p) = \sqrt{ c^2 \vec p^2 + (m c^2)^2 }$	energy-momentum relation